Question

How do u find the formula of the nth term of this arithmetic sequence \[15,{\text{ }}27,{\text{ }}39, \ldots .\] ?

Answer 1

Hint: An arithmetic sequence or arithmetic progression is a series of numbers, in which each term (number) is obtained by adding a fixed number to its preceding term. Here, we are given an arithmetic sequence \[15,{\text{ }}27,{\text{ }}39, \ldots .\] . We need to find the formula of the nth derivative for this arithmetic sequence. Thus, the value of the first term, second term, and so on. Then, find the difference and see the common difference. After using this, we will get the final output.

Complete step-by-step answer:
A sequence is a list of items/objects which have been arranged in a sequential way.
Given that, an arithmetic sequence as: \[15,{\text{ }}27,{\text{ }}39, \ldots .\]
So, the first term is: \[{a_1} = 15\]
Then, the second term is: \[{a_2} = 27\]
The third term is: \[{a_3} = 39\]
Now, the difference between the second and first terms is:
\[ = 27 - 15\]
\[ = 12\]
Also, the difference between the third and second terms is:
\[ = 39 - 27\]
\[ = 12\]
Thus, the common difference is: \[d = 12\]
Therefore, using the above explanation, we can say that, the nth term of the arithmetic sequence is given by the formula: \[{a_n} = a + (n - 1)d\]
So, for \[{a_n}\], the sequence will be as:
\[\therefore {a_n} = a + (n - 1)d\]
Substitute the values of a and d, we will get,
\[ \Rightarrow {a_n} = 15 + (n - 1)12\]
\[ \Rightarrow {a_n} = 15 + (12n - 12)\]
Removing the brackets, we will get,
\[ \Rightarrow {a_n} = 15 + 12n - 12\]
\[ \Rightarrow {a_n} = 15 - 12 + 12n\]
On simplifying this, we will get,
\[ \Rightarrow {a_n} = 3 + 12n\]
So, the correct answer is “\[{a_n} = a + (n - 1)d\]”.

Note: An itemized collection of elements in which repetitions of any sort are allowed is known as a sequence, whereas series is the sum of all elements. A sequence in which every term is created by adding or subtracting a definite number to the preceding number is an arithmetic sequence. The types of sequence and series are:
1) Arithmetic sequences
2) Geometric sequences
3) Harmonic sequences
4) Fibonacci numbers